Sunday, April 17, 2011

After briefly starting to optimize triangle counts for textures in MagicTile, I had a fun realization. The triangle patterns sparked the idea that there could be a precise hyperbolic analogue to a geodesic dome. I was compelled into the diversion, and with minor code changes made some pretty pictures of "geodesic saddles". (That seems like a nice name for these objects anyway.) Alas, my intended optimizations are yet to be done, but at least I can present this geodesic saddle based on the {3,7} tiling :)

Can you find some of the "knots"? That may not be the proper term, but I mean those rare points in the saddle where seven triangles meet at a vertex instead of six. On a geodesic dome, which is usually based on the spherical {3,5} tiling (aka icosahedron), the analogous points are the rare vertices where five triangles meet instead of six. I find knots easier to spot on a geodesic saddle derived from a {3,9} tiling.

Geodesic domes and saddles are generated by taking the tiles in a triangular tiling and subdividing each of them into smaller triangles. Hence, triangular numbers make a cameo in the calculations. For these pictures, I chose to subdivide the original triangles with eight new triangles per side.

But there was one thing that tripped me up quite a bit. I began by mistakenly thinking I could subdivide the triangle edges of the original tiling equally, and then interpolate interior points thereafter. As much as I tried, things just wouldn't line up quite right, and I wasn't seeing the geodesics that I expected. It turns out that all the small triangle edges have varying lengths, something that is also true for a geodesic dome. Compare the proper {3,9} geodesic saddle above with the waviness of an incorrect effort.

I've been showing these pictures in the Poincare Disk, and I don't have unprojected renderings at the moment. But despite this, one thing is certain - a portion of geodesic saddle would make for a unique and fantastic jungle gym!